International Journal of Mechanical Sciences 111-112 (2016) 125–133

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International Journal of Mechanical Sciences

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Plastic collapse of cylindrical shell-plate periodic honeycombs underuniaxial compression: experimental and numerical analyses

Qiang Chen a,n, Quan Shi a, Stefano Signetti b, Fangfang Sun c, Zhiyong Li a, Feipeng Zhu c,Siyuan He a, Nicola M. Pugno b,d,e,nn

a Biomechanics Laboratory, School of Biological Science and Medical Engineering, Southeast University, 210096 Nanjing, PR Chinab Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, viaMesiano 77, I-38123 Trento, Italyc School of Mechanics and Materials, Hohai University, Nanjing 210098, PR Chinad Centre for Materials and Microsystems, Fondazione Bruno Kessler, via Sommarive 18, I-38123 Povo (Trento), Italye School of Engineering and Materials Science, Queen Mary University of London, Mile End Road, E14NS London, UK

a r t i c l e i n f o

Article history:Received 5 November 2015Received in revised form21 March 2016Accepted 23 March 2016Available online 30 March 2016

Keywords:Cylindrical shell-plate honeycombLow-density honeycomb structureFinite element analysisPlastic collapse

x.doi.org/10.1016/j.ijmecsci.2016.03.02003/& 2016 Elsevier Ltd. All rights reserved.

esponding author. Tel./fax: þ86 25 83792620responding author at: Laboratory of Bio-Inics, Department of Civil, Environmental and Mof Trento, via Mesiano 77, I-38123 Trento,9 0461282599.ail addresses: chenq999@gmail.com (Q. Chen)ugno@unitn.it (N.M. Pugno).

a b s t r a c t

This paper studies the plastic collapse mechanisms of uniaxially-loaded cylindrical shell-plate periodichoneycombs with identical mass (or relative density) but varying geometric parameters, by series of in-plane and out-of-plane experiments and finite element numerical simulations. The coupled experi-mental-numerical results show that mechanical properties of the honeycomb can be optimized in allthree loading cases, thanks to the complementary changes of the mechanical properties of cylindricalshell and plate as the geometric parameters vary. The work presents a concept to optimize latticestructures by combining different substructures, and can be used in designing new low-density hon-eycomb structures with desired mechanical requirements but less base materials and weight.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Honeycomb and foamy structures widely exist in natural ma-terials (e.g., Luffa sponge [1]), which are shaped by their sur-rounding environment. In view of optimization in the process ofnatural evolution, it facilitates the study of the natural or man-made foamy materials with multiple functions, such as energy-absorption [2], heat-transfer [3,4] and electromagnetic absorption[5]. To date, varieties of honeycomb structures are presented. Fromthe point of view of mechanics, the conventional hexagon, square,triangle, Kagome honeycombs [6] or 3D topological lattices [7]have been widely studied, and their collapse mechanisms clearlydescribed [8–10]. These studies have already been used as gui-dance for the design of porous materials.

As the importance of hierarchical strategy in natural materialsis gradually realized, hierarchical honeycomb and foamy

.spired and Graphene Nano-echanical Engineering, Uni-

Italy. Tel.: þ39 0461282525;

,

structures are also constructed and studied. In general, as theporosity of the structure increases, reducing the relative density ofthe structure, their mechanical properties decrease (e.g., criticalbuckling load [11]). However, if a hierarchical structure is designedwith a constant amount of bulk materials, its mechanical proper-ties, such as Young's modulus and strength [12], can be optimizedby replacing solid cell walls [13–15] or joints [16,17] of conven-tional single-level honeycombs with porous substructures. In thecase of the solid cell walls replaced by porous substructures, thebending rigidity of the cell walls is increased because of the in-creased porous cell-wall thickness [12,18], and in the other case,i.e., joints replaced by porous substructures, the failure modes ofthe cell walls and porous joints are interchanged [19]. It is worthmentioning that when the hierarchical level comes down into thenanoscale, the surfaces effect has to be taken into account [20,21].

To look for new and more efficient honeycomb structures, veryrecently, the authors presented a cylindrical shell-plate assembledperiodic (or so-called hollow-cylindrical-joint) honeycomb (Fig. 1)[19], which actually derives from the family of center-symmetricalhoneycombs [22], and analytically studied its Young's modulus,Poisson's ratio, fracture strength and toughness in the x direction.Moreover, they reported that the mechanical properties of thestructure were optimized and improved compared to the con-ventional hexagonal honeycombs, thanks to the variation of

Q. Chen et al. / International Journal of Mechanical Sciences 111-112 (2016) 125–133126

structural styles. But the work only dealt with the in-plane me-chanical properties in the x direction via a theoretical approach[19]. Then, questions rise: what about the in-plane mechanicalproperties in the y direction and out-of-plane mechanical prop-erties? Are they also optimized?

In order to thoroughly describe the mechanical behaviours ofthe assembled periodic honeycomb, we here continued the studyof the honeycomb by coupling experiments and numerical simu-lations. First, two sets of quasi-static crushing experiments (six in-plane samples with eight unit cells loaded in the x, y directions,Fig. 1a, and five out-of-plane samples with three unit cells in the z

Fig. 1. Images of (a) an in-plane loaded sample and (b) an out-of-plane loadedsample.

Table 1Geometric parameters and masses (m) of the samples. The numbers in the brackets de

In-plane loaded sam

r/l n. l [mm] r [mm] t [mm]

0.0 — — — —

0.2 1 20 4 1.26

0.3 2 20 6 1.08

0.4 3 20 8 0.95

0.5 — — — —

direction, Fig. 1b) were performed. Plastic collapse processes ofselected samples and stress–strain curves of all samples were re-corded. Second, non-linear finite element method (FEM) numer-ical simulations were employed to deeply reveal plastic and frac-ture behaviours and stress states, which were not visually identi-fiable with the experiments. Finally, the collapse mechanismswere discussed.

2. Methods

2.1. Experiments

Using 6061-T4 aluminum alloy as the bulk constituent material,eleven honeycomb samples with a controlled dimensional error0.03 mm in thickness, were fabricated by Nanjing Siyou Photo-electric Technology Limited, Nanjing, China.

In the design of the samples, the samples had theoreticallyidentical relative density ρ* ρ =/ 0.1c , in which ρ* = 0.27 g/cm3 andρ = 2.7c g/cm3 are the densities of the honeycomb and the bulkaluminum alloy, respectively. The sizes in z direction were 20 mmand 30 mm for the in-plane and out-of-plane samples, respectively,and the distance l between centres of two adjacent cylindrical shellswas fixed to be 20 mm. The radius r of the cylindrical shell was firstselected, and the wall thickness t was determined by the followingequation ρ* ρ = − ⋅( ) + [ ⋅( ) + ]⋅( )t l r l t l/ 1.155 / 2.528 / 1.155 /c 2 [19], andthe masses of the models were calculated by multiplying the vo-lumes of the samples to their density ρ* = 0.27, see Table 1.

Actually, their real masses mre are lower, see Table 1. Moreover,according to the design, the masses of the in-plane samples shouldtheoretically be equal, and also for the out-of-plane ones, here themass difference is due to the samples’ processing.

The samples were tested under uniaxial compression with a1000HDX Instron Universal Testing Machine (ITW, USA) withloading capacity of 1000 kN. Before testing, in order to ensure thesamples to be loaded uniformly, two steel plates were respectivelyplaced at the top and bottom surfaces of the samples. The wholeloading process was displacement controlled from the bottom up.For the in-plane samples, considering their larger size along theloading direction, a short linear-elastic stage of stress–straincurves and limit influence of loading rate on the plastic collapsestage were expected; thus, the loading rates before and afterthe initial yield of the samples were set to be 1 mm/min and10 mm/min, respectively. For the out-of-plane samples, the load-ing rate was kept constant at 1 mm/min.

Regarding the definitions of stress and strain of the compressedhoneycombs in the experiments, the stress was calculated as

note the masses of the samples compressed in the y direction.

ples Out-of-plane loaded samples

m [g] mre [g] n. l [mm] r [mm] t [mm] m [g] mre [g]

— — 1 20 0 1.78 41.7 39.659.9 55.8

(53.1)2 20 4 1.26 41.7 38.9

59.9 55.0(52.3)

3 20 6 1.08 41.7 39.8

59.9 56.7(53.2)

4 20 8 0.95 41.7 36.5

— — 5 20 10 0.84 41.7 34.6

Fig. 2. Stress-strain curve of 6061-T4 aluminum alloy used for the fabrication ofhoneycombs, obtained from tensile test on a dog bone specimen (picture of thefailed specimen depicted within the graph).

Q. Chen et al. / International Journal of Mechanical Sciences 111-112 (2016) 125–133 127

s¼F/A, in which F was the applied load, and A was the projectedconvex hull area of the honeycomb samples on the plane per-pendicular to the loading direction; the strain was calculated asε¼Δh/h0, where Δh was the height variation, and h0 was theinitial height of the compressed samples.

For the numerical models, material properties and stress–strainrelationship of the aluminum alloy used in the experiment werecharacterized by tensioning a dog-bone specimen with circularcross-section of diameter d¼10 mm up to failure (see the inset inFig. 2). The mechanical properties extracted from the stress–straincurve (Fig. 2) were: Young's modulus E¼68 GPa, the yield strengthsy¼287 MPa, the peak stress su¼318 MPa, and the failure strainεf¼0.121. These values are consistent with those in literature [23].

2.2. Finite element models

The aim of finite element simulations is to analyze the stressstate of the honeycombs under compression loads, to visualize theplastic deformation, and to further to develop a referential tool forfuture mechanical analysis of other honeycombs.

The FEM model geometry was made up of honeycombs be-tween two rigid steel plates. The honeycomb material was mod-eled with a piecewise elastic-plastic curve: the linear elastic stagewas defined by the experimentally determined sy and E, while theplastic stage followed the actual experimental points (Fig. 2) inorder to include the softening stage after the reach of the peakstress prior to failure. It is worth mentioning that due to the size-scale effect, the plastic strain εpl,FEM and ultimate strain εu,FEM inthe FEM models were both scaled from the nominal one εpl

measured from the dog-bone test by ε ε = d t/ /pl,FEM pl , since t«d. Thisscaling was necessary, since by introducing the material law de-rived from dog-bone traction in the FEM material model we ob-tained brittle fracture at low strain, while with this scaling wewere able to obtain comparable results with the experimentalcompression tests. Von-Mises criterion was employed for yielding.Contact interactions were taken into account between the steelplates and the honeycomb and self-contact within the honeycombparts in the crushing process: static and dynamic coefficients offriction were respectively set to be 0.61/0.47 for the honeycomb-steel contact and 1.35/1.05 for the self-contact. For the in-planeand out-of-plane models, 4-node fully integrated shell elementswith 2�2 Gauss integration (4 integration points through thick-ness, element side-thickness aspect-ratio 2:1) and 8-node selec-tive reduced integrated brick elements (3 elements in the wall

thickness, aspect ratio 1:1:1) with volumetric locking alleviationwere employed, respectively. The loading process was displace-ment controlled with the bottom one moved towards the top atthe same prescribed velocities as the experiments reported above.The load F carried by the honeycomb was computed from theresultant component along the loading axis of the contact forces atthe steel plate-honeycomb interface. Material fracture was treatedvia an erosion algorithm: when all the integration points of anelement reach the ultimate strain εu,FEM (maximum principalstrain), the element is deleted from simulation, and then fractureenucleate and propagate by progression of erosions.

3. Results

3.1. In-plane mechanical behaviour

The in-plane mechanical behaviour of the six honeycombscompressed in the x and y directions is plotted in Fig. 3. It can beseen that the stress–strain curves generally take on a serratedfeature, due to the fracture in cylindrical shells and plates, and thelinear-elastic stage (E1) of all samples is very short. The FEM re-sults (dashed line) and experimental results (solid line) are in goodagreement, both in terms of curve shape and of honeycombbearing capacity. The samples 1 and 3 (r/l¼0.2, red line andr/l¼0.4, blue lines) have only one plateau stage, whereas, thesample 2 (r/l¼0.3, green line) has two linear-elastic or plateaustages (Fig. 3a and b), and the second linear-elastic (E2) and pla-teau (P2) stages are longer and much shorter than their firstcounterparts (i.e., E1 and P1), respectively. The sample 2 reachesdensification earlier than the samples 1 and 3.

Young's moduli (the slope of linear-elastic stages, see the insetsin Fig. 3a and b) of the honeycombs are optimized in both direc-tions when r/l¼0.3 for FEM and experimental results, but thevalues from FEM result are much larger than those from experi-ments, and this is consistent with the literature [24,25]. It is ana-lyzed that the discrepancy is mainly due to the over-estimatedstrain, which was calculated by employing the cross-head dis-placement of the testing system instead of the real deformation ofthe samples. However, because this work studies the plastic col-lapse behaviour of the honeycomb, Young's modulus is not in thescope.

Yield strengths of the three samples in the x direction are1.11 MPa (r/l¼0.2), 1.20 MPa (r/l¼0.3), 1.02 MPa (r/l¼0.4) for ex-periments vs. 1.19 MPa, 1.29 MPa, 1.10 MPa for FEM; in the y di-rection, and they are 1.19 MPa (r/l¼0.2), 1.34 MPa (r/l¼0.3),0.96 MPa (r/l¼0.4) for experiments vs. 1.28 MPa, 1.42 MPa,1.10 MPa for FEM. Differently from the Young's modulus, FEM andexperimental values are comparable since the stress in experi-ments calculated directly from the readout of the load cell, whichreflects the real load sustained by the samples. From these data,we can see that the yield strengths are optimized when r/l¼0.3,which is consistent with the previous work [19], and the finiteelement results are slightly greater than the experimental coun-terparts. Moreover, the yield strength in the x direction is less thanthat in the y direction, and this is caused by the different struc-tures of the two directions, with the compressed samples in the xdirection including the extra axial deformation or instability of thevertical plates. Considering the effect of the mass variations amongsamples, we here also compare the yield strength to mass ratio,and again the optimal case for both directions corresponds tor/l¼0.3 (Fig. 3c and d). The coherent optimizations (yield strengthand yield strength to mass ratio) are due to the little mass dif-ference for each intra-group (i.e., two groups are x and ydirections).

In addition, the large deformations and failure mechanisms

Fig. 3. Stress-strain curves of the sample loaded in (a) x direction and (b) y directions showing experimental (solid lines) and FEM (dashed lines) results. Yield strenght tomass ratio for (c) x direction and (d) y direction with comparison between experiments (filled markers) and FEM (unfilled markers connected by dashed lines). (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Q. Chen et al. / International Journal of Mechanical Sciences 111-112 (2016) 125–133128

(Figs. 4 and 5) in both directions are observed. For the x direction,the samples 1 and 2 (i.e., r/l¼0.2, 0.3) have an approximate anti-symmetric configuration (Fig. 4a and c), thanks to the instabilityof vertical plates, which have a relative large slenderness ratio,while the sample 3 is nearly symmetric (Fig. 4e). When thesamples fail, the plastic hinges and fracture points locate at theplate-shell connecting points on the plates (points A and B,Fig. 4g) or on the cylindrical shell (points C, Fig. 4h), dependingon the r/l ratio. This behaviour has already been verified by thetheoretical analysis [19]. For the y direction, the three samplesshare a symmetric configuration (Fig. 5a,c, and e), but their fail-ures differ as well, since in the samples 1 and 2 (i.e., r/l¼0.2, 0.3)plastic hinges and fractures occur in the two plates (points A inFig. 5g), while on the cylindrical shell in the the sample 3(r/l¼0.4, points C in Fig. 5h). The same occurs in the samplesloaded in the x direction. The contours of von Mises stresses fromthe finite element simulations confirm the results, see Figs. 4i and5i.

3.2. Out-of-plane mechanical behaviour

Like the in-plane case, the FEM and experimental stress–straincurves and the yield strength to mass ratio of the five out-of-planesamples are plotted in Fig. 6. In Fig. 6a, the portions (marked by I, IIand III) between elastic limit and yield stress of the strain–stresscurves of the sample 2, 3, 4 is longer than those of the samples1 and 5. This is because in the loading process, the three samplesunderwent different but slight shear effect, which reduced the

samples' yield strength. Moreover, when r/l is small (samples 1–3),there is only one couple of peak and valley; while r/l is large(samples 4, 5), there are multi-couples of peak and valley (①�④),and the plateau stage is much longer than those of the sampleswith small r/l, but sample 3 (r/l¼0.3) still possesses the greatestenergy-absorption capacity (computed as the area under thestress-strain curve). And more, it is readily seen that there arejumps (dashed squares in Fig. 6a) in the strain–stress curves forsamples 2 and 4, and this is due to the cell-wall brittle fracture.Besides, the FEM compression tests are shown in Fig. 6a1–a3.

Young's moduli reflected by the slopes of the linear-elasticstages are approximately same (Fig. 6a), and this can be explainedthrough the classical predication by * = ρ* ρE E/ /c c [10], and hereρ* ρ/ c is a constant. Young's moduli from FEM result are much largerthan those from experiments, and the reason can be referred tothe in-plane cases.

The yield strengths of the five samples are 35.95 MPa (r/l¼0.0),35.46 MPa (r/l¼0.2), 38.78 MPa (r/l¼0.3), 30.60 MPa (r/l¼0.4),27.81 MPa (r/l¼0.5) for experiments vs. 34.99 MPa, 34.51 MPa,38.06 MPa, 29.09 MPa, 26.80 MPa for finite element simulations.The yield strength of the out-of-plane samples is 20–30 timesthose of the in-plane samples. The optimized yield strength isobtained as well when r/l¼0.3, and the same for the yieldstrength to mass ratio (Fig. 6b) due to the weak mass variation ofthe intra-group (out-of-plane) samples. However, different fromthe in-plane samples, the FEM results are lower than their ex-perimental counterparts.

As an interesting example, we snapshotted four peak-valley states

Fig. 4. Snapshots of the experimental in-plane loaded samples in the x direction with r/l¼0.2 (a and b) r/l¼0.3 (c and d) r/l¼0.4 (e and f) at two different strain levels.(i) Corresponding snapshots from finite element simulations and details of the cylindrical shell-plate joints for different r/l with contour of von Mises stress (red regions arethe most stressed). The solid coloured circles in (g and h) represent the plastic hinges or fracture locations corresponding to the empty circles in (b, d and f). (For inter-pretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Q. Chen et al. / International Journal of Mechanical Sciences 111-112 (2016) 125–133 129

of the sample 5 assembled only by cylindrical shells without plates,and the sequence of events is shown in Fig. 7. Initially, the entiresample deforms elastically to the first peak point, i.e., yield stress. Dueto the existence of the bottom steel plate, the further expansion of thesample's deformation is restrained, and the first axisymmetric out-ward fold (n¼1) starts to form; as the load increases, the fold of eachcylindrical shell grows and it thrusts into its adjacent shells to form anoverlap (the arrows in ① of Fig. 7), while the rest of the sample stilldeforms elastically. Simultaneously, the constituent material at the foldbegins to yield, and the entire sample shows a softening behaviour.Then, the deformation of the portion close to the fold accumulates.After the contact of the two sides in the fold, the drop of the com-pressive load arrests. Meanwhile, a new diamond fold (n¼2) starts toform, and the second peak stress gradually emerges, followed by thesecond valley. In state ②, the overlaps (the arrows in ② of Fig. 7) canbe clearly seen. After the state ②, the third and fourth diamond folds(states ③ and ④) are formed by the squeezed cylindrical shells (thearrows in ③ and ④ of Fig. 7), and the stress does not apparently in-crease until it reaches the densification of state ④, after which, thestress increases sharply.

4. Discussions

To date, a number of lattice structures have been studied anddefined [7]. Here, the structure is discussed in a new sense, i.e., it is

regarded as a periodic combination of plate and cylindrical shells,which are two basic elements in structural mechanics. As r/l rises,the contribution of the plate decreases but the one of cylindricalshell increases, and these complementary tendencies optimize themechanical behaviour of the structure. Not losing generality, weconsider this as an optimized spatial arrangement of n sub-structures. The general mechanical behaviour F of the structure isexpressed as the sum of the n substructures

∑= ( )( )=

F f M S P, ,1i

n

1i i i i

where fi represents the mechanical contribution of the i-thsubstructure, Mi, Si and Pi are mechanics-parametric, size andpositional information of the i-th substructure, respectively. Thegeneral mechanical behaviour F in Eq. (1) can be the force-dis-placement curve, elastic modulus, strength, toughness, and othermechanical properties. As we know, for a structure, the selectionof bulk materials (i.e., Mi) could be referred to Ashby's maps [10],which provide materials indices for mechanical designs accordingto stiffness, strength and other variables. Then, mechanical prop-erties of the structure can be obtained by varying the size (Si) andspatial arrangement (Pi) of the substructures. Therefore, given aconstituent material, for the optimization, we have

Fig. 5. Snapshots of the experimental in-plane loaded samples in the y direction with r/l¼0.2 (a and b) r/l¼0.3 (c and d) r/l¼0.4 (e and f) at two different strain levels.(i) Corresponding snapshots from finite element simulations and details of the cylindrical shell-plate joints for different r/l with contour of von Mises stress. The solidcoloured circles in (g and h) represent the plastic hinge or fracture locations. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

Q. Chen et al. / International Journal of Mechanical Sciences 111-112 (2016) 125–133130

∂∂

=( )

FS

02ai

and

∂∂

=( )

FP

02bi

The present structure, with the specific relative density (i.e.,ρ* ρ/ c¼0.1), is a peculiar case of the above optimized solution withthe 6061-T4 Aluminum-alloy material (Mi), fixed positions of theplate and cylindrical shell (Pi), and two substructures (n¼2). Thus,the only factor to be optimized is the size variable r/l, and theoptimized value is about 1/3. Considering Eq. (1), we find thesimple expression

= + ( ≠ = )( )

Ff

f

fi j i j P CS1 ; , or

3i

j

i

where fi and fj are functions of the honeycomb's relative density

ρ* ρ =/ 0.1c and of the size variable r l/ , and they denote the me-chanical behaviours of plates (P) or cylindrical shells (CS), re-spectively. For a specific case, in the linear-elastic stage, we

consider F, fP and fCS as the bending elastic strain energies of thewhole structure, plates and cylindrical shells, respectively. Ac-cording to the calculation of the strain energy by the authors'previous work [19], the competitive relation of the two parts isplotted in Fig. 8, from which we can clearly see that the con-tributions of plates and shells in the linear-elastic deformation ofthe honeycomb when r/l varies.

4.1. In-plane loaded samples

We have shown the mechanical behaviour of the six samples inSection 3.1. The feature is that the samples 2 (both x and y di-rections) have two different linear-elastic and plateau stages inexperiments. According to Eq. (3), for the two extreme cases, ifr/l-0, the cylindrical shells disappear, i.e., =f 0CS , and the struc-ture shrinks into the conventional regular hexagonal honeycomb,which has been widely studied in literature. Thus, we have =F fP;otherwise, when r/l-0.5, the plates disappear, i.e., =F fCS. For thestructures in-between, smaller r/l ratio (e.g., sample 1) results in amore rigid cylindrical shells, and the in-plane samples fail in theweaker plates due to bending and buckling; on the contrary,greater r/l (e.g., sample 3) results in a more compliant cylindrical

Fig. 6. Comparison between experimental (solid lines) and finite element simulations (dashed lines) results of out-of-plane loaded samples. (a) Strain–stress curves of thefive samples, and snapshots from the finite element simulations of the optimized structure at three states and the contour of plastic strain of the second state and (b) yieldstrength to mass ratios of the five samples: filled and dashed-line connected unfilled spots represents the experimental and FEM results, respectively. Note: the sample 1 (r/l¼0.0) denotes the conventional regular hexagonal honeycomb.

Fig. 7. Snapshots of four folds in the loaded sample 5: (a) experiment and (b) finite element simulation.

Q. Chen et al. / International Journal of Mechanical Sciences 111-112 (2016) 125–133 131

shell, and the samples fail in the weaker cylindrical shell due totheir bending. These two cases result in single pair of linear elasticand plastic stages as shown in Fig. 3a and b. When fP is compar-able to fCS, the two parts fail one after the other, namely, if <f fP CS,

the plate fails before the cylindrical shell, which is the case ofsample 2. This causes the double pairs of linear elastic and plasticstages, E1-P1 and E2-P2 in Fig. 3a and b: the first one is con-tributed by the bending and yielding of the plate, and the second

Fig. 8. Competition of elastic strain energies between the plate and the cylindricalshell as r/l varies.

Fig. 9. Collapse modes of the (a) sample 1, and (b) sample 5. Simulations show thecontour of plastic strain, and a good agreement in the collapse mechanism withexperimental results. (For interpretation of the references to colour in this figure,the reader is referred to the web version of this article.)

Q. Chen et al. / International Journal of Mechanical Sciences 111-112 (2016) 125–133132

by the bending and collapse of the cylindrical shell. In view of this,it can be concluded that it is the featured structure that providesthe sample 2 with the optimized failure mechanism and furtherbest energy-absorption ability even though the six samples haveapproximate mass. In this regard, the one-after-another failuremechanism of different components in the structure is similar tothe hierarchical behaviour of spider silk [26], which enables itsgreat extensibility, toughness and strength.

4.2. Out-of-plane loaded samples

In the loading process, the samples 1 and 5 were compressedalmost aligned with the centroid of the imprint area of the twosamples. The collapsed samples and their failure mechanisms areshown in Fig. 9. For the sample 1, the collapse of the structure,caused by the plate, is a classical problem, and the plasticstrength is simply predicted by

σ*σ

= (λ φ) ρ*ρ ( )

⎛⎝⎜

⎞⎠⎟C ,

4ysc c

2

where, σ* and σcys are yield strengths of the structure and its con-

stituent material, and C is a constant, which depends on the lobe'swavelength λ and the rotational angle ϕ of the plastic hinge. Theblack circled part (upper one in Fig. 9a) is representative because itsboundary condition is close to that of the unit cell in multi-cellstructures (only three unit cells here). The failed plates exhibit ananticlockwise trichiral arrangement, and only one lobe and fracturemouth forms (highlighted by the arrow in Fig. 9a), which dependson the structural geometry and mechanical behaviour of its con-stituent materials.

For the sample 5, the collapse of the structure is caused by thecylindrical shell. For single cylindrical tube, many works explainits collapse mechanisms [27–29]. In particular, depending onthickness, diameter and length, Andrews et al. [29] classifiedcollapse modes of cylindrical tubes into seven groups from con-certina to tilting of the tube axis, basing on tested 189 annealedHt-30 Aluminium alloy tubes. However, the cylindrical shell inthe present structure is different from the single cylindrical tube,because of the restrains of its adjacent cylindrical shells (redcircled points in Fig. 9b). In this case, the three-lobe collapse isprone to occur. Besides, we find that the sample 1 has a largerfold size and less fold number than those of the sample 5.

For the samples 2–4, their collapse is contributed by both parts,

i.e., plate and cylindrical shell (Fig. 10). Combining the extremecase (r/l¼0 and r/l¼0.5) discussed above, we can conclude that asr/l increases (Fig. 10), the substructure dominating the collapse ofhoneycombs again changes from the plates to cylindrical shells, aspredicted by Eq. (3). The numbers of folds in plates and cylindricalshells are mutually dependent, see the arrows in Fig. 10d, and thefold number increases for increasing r/l, and correspondingly, theirwavelength decreases. Finally, due to the instability of the platesand cylindrical shells, marked by ellipses in Fig. 10a–c, the sheareffect as stated before is introduced, and this corresponds to thethree stages (I, II, III) in Fig. 6a.

5. Conclusions

In this paper, we studied the mechanical behaviour of a set ofcylindrical shell-plate assembled honeycombs. Through experi-ments and FEM numerical simulations carried on in-plane andout-of-plane loaded samples, their plastic collapse mechanismswere observed and analyzed. Young's modulus and yield strengthof the honeycombs were optimized when r/l¼0.3. In particular,when r/l¼0.3, the in-plane loaded samples exhibited a functionalgradient property since the plates failed after the cylindrical shells,while the out-of-plane direction was associated to the best en-ergy-absorption capacity. Meanwhile, the developed numericalmodel was verified to be able to describe the experiments andcould be used in the studies of future-developed honeycombs.Most of all, it indicates that the combination of hollow cylindricalshells and plates forms a new periodic assembly with better me-chanical properties with respect to conventional honeycombs. Thisstrategy may be used to generate new lattices for enhancedcrashworthy structures.

Fig. 10. Experimental and simulated collapse modes of the out-of-plane loaded honeycombs with (a) r/l¼0.2, (b) r/l¼0.3, (c) r/l¼0.4 and (d) collapse mechanisms of thecylindrical shell-plate joints observed in the simulations, with detail of wall folding.

Q. Chen et al. / International Journal of Mechanical Sciences 111-112 (2016) 125–133 133

Acknowledgements

This work was partially supported by the National Natural Sci-ence Foundation of China (NSFC) (Nos. 31300780, 11272091, and11572087, 11572087) and the Fundamental Research Funds for theCentral Universities (No. 2242016R30014). NMP is supported by theEuropean Research Council (ERC StG Ideas 2011 BIHSNAM n.279985, ERC PoC 2013 KNOTOUGH n. 632277, ERC PoC 2015 SILK-ENE n. 693670) and by the European Commission under the Gra-phene Flagship (WP "Nanocomposites", n. 696656). SS acknowl-edges support from BIHSNAM.

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